Introduction to Computational Fluid Dynamics - Numerics - 1 - Finite Difference and Spectral Methods

[@bumpty9830]

Maybe beyond the scope of this course, but my favorite application of the Laplacian operator in this area is mesh smoothing: treating the coordinates as solution variables of a Laplace (or Poisson) equation, residuals can be computed and the coordinates bumped (with sampling) toward the solution.

[@leonmorais2584]

Really good content professor, thank you for sharing!

[@SAEMiller]

You're welcome - next semester I will be posting a course on compressible flow!

[@maosenye423]

I am a master's student in Utokyo. Your course really helps me a lot. Personally, I think it will be better to establish a course website.

[@SAEMiller]

Thanks for your nice compliment! There is a class website but its for enrolled students at the University. I might make a website for the public later. Thanks again! :)

[@brianhowell7626]

excellent explanations

[@SAEMiller]

Glad it was helpful!

[@dr.engineering201]

@@SAEMiller Hi I am trying to derive some of these equations and need your help.

[@roros2512]

Thank you for the great lecture professor Miller. I'd like to ask, the spectral method, is only applicable on structured grids and only using finite difference approach? is it possible to use it on an unstructed mesh? is it possible to use it using finite voume method? Thank you again

[@SAEMiller]

There are a number of new papers appearing called the spectral element method. One of my own papers uses the method. It is being used in many new codes now. Suggest searching for spectral element method on google scholar.

[@husseinkokash1569]

You are simply amazing Professor, a life savior, by any chance; will you in the near future make a detailed class on spectral methods and how it is implemented in CFD analysis?

[@SAEMiller]

There are two major types of spectral methods appearing in contemporary CFD. The first is with unstructured meshes and the second is with respect to traditional structured DNS. Right now, I believe there are great resources within articles on spectral meshes for DNS. There are very few resources on unstructured spectral methods. This is really an emergent area of CFD in the last number of years and is actually a subject of research in my group. Because it's such a niche subject I do not have any plans to make those videos. As its a really great idea - perhaps you can create one! :)

[@zeeshanbashir7428]

Dear Professor your voice muted at certain times in the video. i.e 41 -42 minute time

[@SAEMiller]

Thank you for bringing that to my attention. I'm not sure what is wrong. Something probably happened with my computer at the time. I'll have to go back and look at the original recording and maybe add a transcript at that location. Edit: It probably isn't too much of a big deal because I was talking about a few finer points.

[@parthdevbundela6744]

Hi sir in spectral methods, how we will calculate the value of wave number k, if grid spacing h is non uniform.

[@SAEMiller]

Dear Parth - Wavenumber involves a spatial Fourier transform. Look at the equation for wavenumber as a function of x. Do you see any assumption or restriction regarding the grid point spacing? The answer is no. It seems you are thinking in terms of algorithms that use a constant \Delta x spacing. If you are working in the discrete world, you can write a transform that uses non constant spacing. Alternatively, consider interpolating your data onto a structured grid of your choosing. I think there is a lot of value in writing your own mathematical library and functions once in life, that way you can fully understand what happens when calling routines from say, Matlab, or another similar system.